State-dependent Riccati Equation Technique Research Articles

Design of Nonlinear Attitude Controller for an Unmanned Helicopter Based on Improved State-Dependent Riccati Equation Technique

Design of Nonlinear Attitude Controller for an Unmanned Helicopter Based on Improved State-Dependent Riccati Equation Technique

Application of the State-Dependent Riccati Equation and Kalman Filter Techniques to the Design of a Satellite Control System

Design of Satellite Attitude Control System (ACS) that involves plant uncertainties and large angle manoeuvres following a stringent pointing control, may require new non-linear control techniques in order to have adequate stability, good performance and robustness. In that context, experimental validation of new non-linear control techniques through prototypes is the way to increase confidence in the controller designed. The Space Mechanics and Control Division (DMC) of INPE is constructing a 3-D simulator to supply the conditions for implementing and testing satellite ACS hardware and software. The 3-D simulator can accommodate various satellites components; like sensors, actuators, computers and its respective interface and electronic. Depending on the manoeuvre the 3-D simulator plant can be highly non-linear and if the simulator inertia parameters are not well determined the plant also can present some kind of uncertainty. As a result, controller designed by linear control technique can have its performance and robustness degraded, therefore controllers designed by new non-linear approach must be considered. This paper presents the application of the State-Dependent Riccati Equation (SDRE) method in conjunction with Kalman filter technique to design a controller for the DMC 3-D satellite simulator. The SDRE can be considered as the non-linear counterpart of Linear Quadratic Regulator (LQR) control technique. Initially, a simple comparison between the LQR and SDRE controller is performed. After that, practical applications are presented to address problems like presence of noise in process and measurements and incomplete state information. Kalman filter is considered as state observer to address these issues. The effects of the plant non-linearities and noises (uncertainties) are considered in the performance and robustness of the controller designed by the SDRE and Kalman filter. The 3-D simulator simulink-based model has been developed to perform the simulations examples to investigate the SDRE controller performance using the states estimated by the Kalman filter. Simulations have demonstrated the validity of the proposed approach, once the SDRE controller has presented good stability margin, great performance and robustness.

Application of the State-Dependent Riccati Equation and Kalman Filter Techniques to the Design of a Satellite Control System

Design of Satellite Attitude Control System (ACS) that involves plant uncertainties and large angle manoeuvres following a stringent pointing control, may require new non-linear control techniques in order to have adequate stability, good performance and robustness. In that context, experimental validation of new non-linear control techniques through prototypes is the way to increase confidence in the controller designed. The Space Mechanics and Control Division (DMC) of INPE is con- structing a 3-D simulator to supply the conditions for implementing and testing satellite ACS hardware and software. The 3-D simulator can accommodate various satellites components; like sensors, actuators, computers and its respective interface and electronic. Depending on the manoeuvre the 3-D simulator plant can be highly non-linear and if the simulator inertia parameters are not well determined the plant also can present some kind of uncertainty. As a result, controller designed by linear control technique can have its performance and robustness degraded, therefore controllers designed by new non-linear approach must be considered. This paper presents the application of the State-Dependent Riccati Equation (SDRE) method in conjunction with Kalman filter technique to design a controller for the DMC 3-D satellite simulator. The SDRE can be considered as the non-linear counterpart of Linear Quadratic Regulator (LQR) control technique. Initially, a simple comparison between the LQR and SDRE controller is performed. After that, practical applications are presented to address problems like presence of noise in process and measurements and incomplete state information. Kalman filter is considered as state observer to address these issues. The effects of the plant non-linearities and noises (uncertainties) are considered in the performance and robustness of the controller designed by the SDRE and Kalman filter. The 3-D simulator simulink-based model has been developed to perform the simulations ex- amples to investigate the SDRE controller performance using the states estimated by the Kalman filter. Simulations have demonstrated the validity of the proposed approach, once the SDRE controller has presented good stability margin, great per- formance and robustness.

Load maximization of flexible joint mechanical manipulator using nonlinear optimal controller

In this paper, a closed loop nonlinear optimal control approach is investigated for flexible joint manipulators (FJM). The dynamic load carrying capacity (DLCC) of these manipulators is obtained via this approach. The state-dependent Riccati equation (SDRE) technique is used for solving nonlinear optimal control problem. This method uses special parameterization to develop the nonlinear system to a linear structure having state-dependent coefficient matrices. The Taylor series numerical method is addressed by approximating the solution to the SDRE. Simulations for FJM are provided to illustrate the effectiveness of this approach for designing nonlinear feedback controllers and computing DLCC of these manipulators. Also effects of change in spring factors of manipulator and control parameters on resulted trajectory of manipulator are considered. DLCC is calculated subject to the limits in actuators and tracking accuracy. Finally the results are checked by comparison with experimental results.

Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique

A non-linear attitude control method for a satellite with magnetic torque rods using the state-dependent Riccati equation (SDRE) technique has been developed. The magnetic torque caused by the interaction with the Earth's magnetic field and the magnetic moment of torque rods plays a role of the control torque. The detailed equations of motion for this system are presented using angular velocity and quaternions. The SDRE controller is developed for the non-linear systems which can be formed in pseudo-linear representations using the state-dependent coefficient (SDC) method without linearization procedure. The aim of this control system is to achieve a stable attitude within 5°, and minimize the control effort. The stability regions for the SDRE controlled satellite system are estimated through the investigation of the stability conditions developed for pseudo-linear systems and the application of Lyapunov's theorem. For comparisons, the Linear Quadratic Regulator (LQR) method using the solution of the algebraic Riccati equation (ARE) is also applied to this non-linear system. The performance of the SDRE non-linear control system demonstrates more robustness and stability than the LQR control system when subjected to a wide range of initial conditions.

Solid Launcher Dynamical Analysis and Autopilot Design

The dynamics of a small solid launch vehicle has been investigated. This launcher consists of a liquid upper stage and three fundamental solid rocket boosters aligned in series. During the ascent flight phase, lateral jets and grid fins are adopted by the flight control system to stable the attitude of the launcher. The launcher is a slender and aerodynamically unstable vehicle with sloshing tanks. A complete set of six-degrees-of-freedom dynamic models of the launcher, incorporation its rigid body, aerodynamics, gravity, sloshing, mass change, actuator, and elastic body, is developed. Dynamic analysis results of the structural modes and the bifurcation locus are calculated on the basis of the presented models. This complete set of dynamic models is used in flight control system design. A methodology for employing numerical optimization to develop the attitude filters is presented. The design objectives include attitude tracking accuracy and robust stability with respect to rigid body dynamics, propellant slosh, and flex. Later a control approach is presented for flight control system of the launcher using both State Dependent Riccati Equation (SDRE) method and Fast Output Sampling (FOS) technique. The dynamics and kinematics for attitude stable problem are of typical nonlinear character. SDRE technique has been well applied to this kind of highly nonlinear control problems. But in practice the system states needed in the SDRE method are sometimes difficult to obtain. FOS method, which makes use of only the output samples, is combined with SDRE to accommodate the incomplete system state information. Thus, the control approach is more practical and easy to implement. The resulting autopilot can provide stable control systems for the vehicle.

A Nonlinear Optimal Guidance Law with Terminal Impact Angle Constraint

State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design methods for nonlinear controllers. A nonlinear optimal guidance law with impact angle constraint is derived for planar engagements to attack stationary targets. The guidance problem is formulated as an infinite horizon nonlinear regulator problem whose equilibrium state is zero. It is solved by SDRE technique and the state weight matrix is chosen as a function of time-to-go. Performance of the guidance law is tested numerically with different initial firing conditions for a realistic GPS/INS guided artillery rocket model with low available lateral acceleration. A reasonable launch angle is helpful to reduce the control effort, and it is acquired by trajectory optimization using genetic algorithm. Results show negligible errors for miss-distance and the desired impact angle. The proposed guidance law is a choice for the guided artillery rocket.

Optimal Sliding Mode Controllers for Attitude Stabilization of Flexible Spacecraft

The robust optimal attitude control problem for a flexible spacecraft is considered. Two optimal sliding mode control laws that ensure the exponential convergence of the attitude control system are developed. Integral sliding mode control (ISMC) is applied to combine the first‐order sliding mode with optimal control and is used to control quaternion‐based spacecraft attitude manoeuvres with external disturbances and an uncertainty inertia matrix. For the optimal control part the state‐dependent Riccati equation (SDRE) and optimal Lyapunov techniques are employed to solve the infinite‐time nonlinear optimal control problem. The second method of Lyapunov is used to guarantee the stability of the attitude control system under the action of the proposed control laws. An example of multiaxial attitude manoeuvres is presented and simulation results are included to verify the usefulness of the developed controllers.

A Useful Control Model for Tandem Hot Metal Strip Rolling

The tandem hot metal strip rolling process is a highly complex nonlinear system that presents a difficult control challenge. This challenge is exacerbated by the hostile hot metal rolling environment, which precludes the location of sensors to measure variables that are important for control. In addition, the controller must have a structure that offers simplicity of tuning during commissioning by personnel who usually are unfamiliar with advanced process control techniques. Based on our previous work using a state-dependent Riccati equation technique for control of the tandem cold metal rolling process, it is considered that a similar method might also be useful for control of tandem hot strip rolling. For the hot rolling process, the development of a process model in a form that is suitable for control development is a significant and challenging task. This paper describes our work to expand on an initial portion of this model to develop a comprehensive nonlinear model of this process. Based on simulation results, it is determined that the complete model has the potential to be useful in the development of a viable nonlinear control method for significant improvement in the control of the tandem hot rolling process.

Satellite formation reconfiguration and station-keeping using state-dependent Riccati equation technique

The current paper presents optimal reconfigurations and formation-keeping for formation flying satellites. The state-dependent Riccati equation (SDRE) technique is utilized as a non-linear controller for both the reconfiguration problem and formation-keeping problem. For the SDRE controller, a state-dependent coefficient (SDC) form is formulated to include non-linearities in the relative dynamics and J 2 orbital perturbation. The Taylor series and a transformation matrix are used to establish the SDC form. Optimal reconfiguration trajectories that minimize energy in satellite formation flying are obtained by the SDRE controller and compared with those obtained from a linear quadratic regulator (LQR) and a linear parameter varying (LPV) control method. It is illustrated that the SDRE non-linear controller of the current study obtains relocation accuracy of less than 0.1% of formation base-line length, while the LQR controller and LPV controller yield relatively large relocation errors. The formation-keeping controller developed using the SDRE technique in the current study also provides robustness under severe orbital perturbations.

Hardware-in-the-loop simulations of GPS-based navigation and control for satellite formation flying

A relative navigation and formation control algorithm for satellite formation flying was developed, and a hardware-in-the-loop (HIL) simulation testbed was established and configured to evaluate this algorithm. The algorithm presented is a relative navigation estimation algorithm using double-difference carrier-phase and single-difference code measurements based on the extended Kalman filter (EKF). In addition, a state-dependent Riccati equation (SDRE) technique is utilized as a nonlinear controller for the formation control problem. The state-dependent coefficient (SDC) form is formulated to include nonlinearities in the relative dynamics. To evaluate the relative navigation and control algorithms developed, a closed-loop HIL testbed is configured. To demonstrate the performance of the testbed, a test formation flying scenario comprising formation acquisition and keeping in a low earth orbit (LEO) has been established. The relative navigation results from the closed-loop simulations show that a 3D RMS of 0.07 m can be achieved for position accuracy. The targeted leader–follower formation flying in the along-track separation of 100 m was maintained with a mean position error of approximately 0.2 m and a standard deviation of 0.9 m. The simulation results show that the HIL testbed is capable of successful demonstration of the GPS-based satellite autonomous formation flying mission.

Systematic and effective design of nonlinear feedback controllers via the state-dependent Riccati equation (SDRE) method

Since the 1990s, state-dependent Riccati equation (SDRE) strategies have emerged as general design methods that provide a systematic and effective means of designing nonlinear controllers, observers and filters. These methods overcome many of the difficulties and shortcomings of existing methodologies, and deliver computationally simple algorithms that have been highly effective in a variety of practical and meaningful applications in very diverse fields of study. These include missiles, aircraft, unmanned aerial vehicles, satellites and spacecraft, ships, autonomous underwater vehicles, automotive systems, biomedical systems, process control, and robotics, along with various benchmark problems, as well as nonlinear systems exhibiting several interesting phenomena such as parasitic effects of friction and backlash, unstable nonminimum-phase dynamics, time-delay, vibration and chaotic behavior. SDRE controllers, in particular, have become very popular within the control community, providing attractive stability, optimality, robustness and computational properties, making real-time implementation in feedback form feasible. However, despite a documented history of SDRE control in the literature, there is a significant lack of theoretical justification for logical choices of the design matrices, which have depended on intuitive rules of thumb and extensive simulation for evaluation and performance. In this paper, the capabilities and design flexibility of SDRE control are emphasized, addressing the issues on systematic selection of the design matrices and going into detail concerning the art of systematically carrying out an effective SDRE design for systems that both do and do not conform to the basic structure and conditions required by the method. Several situations that prevent the direct application of the SDRE technique, such as the presence of control and state constraints, are addressed, demonstrating how these situations can be readily handled using the method. In order to provide a clear understanding of the proposed methods, systematic and effective design tools of SDRE control are illustrated on a single-inverted pendulum nonlinear benchmark problem and a practical application problem of optimally administering chemotherapy in cancer treatment. Lastly, real-time implementation aspects are discussed with relevance to practical applicability.

A New Strategy for Optimal Control of Continuous Tandem Cold Metal Rolling

The control of continuous tandem cold rolling of metal strip is similar to the control of stand-alone tandem cold rolling except that additional challenges occur during the passage of the weld, which generally is done at reduced speed. The more significant issues that must be considered are the following: 1) reducing the length of strip near the weld that has excessive excursions in thickness; 2) reducing the excursions in tension and roll force as the weld goes through a stand; and 3) maintaining the mass flow balance in the mill. This paper presents the results of an investigation of the application of the state-dependent Riccati equation (SDRE) technique developed for improvement in the control of stand-alone tandem cold rolling, to continuous tandem cold rolling, particularly during passage of the weld. Two methods of control during this regime of operation are evaluated and a preferred method is selected. Using the preferred method, it was determined by simulation that the SDRE technique has the capability for successfully controlling the mill during weld passage, so that this novel approach offers a strong potential for improvement in the control of both the stand-alone and continuous tandem cold rolling processes.

An Initial Model for Control of a Tandem Hot Metal Strip Rolling Process

The tandem rolling of hot metal strip presents a significant control challenge because of nonlinearities and process complexities. The challenge is heightened by an extremely hostile environment that precludes the location of sensors to measure process variables that are important for control. In addition, it is essential that the controller structure allows for a high degree of physical intuition in the design process and provides for simplicity of tuning during commissioning. Based on our previous work using a state-dependent Riccati equation (SDRE) technique for control of the tandem cold rolling process, it is considered that a similar method might also be useful as a basis for the development of a control technique for tandem hot strip rolling. For the hot rolling process, the development of a process model in a form that is suitable for use with a SDRE-based method is a significant and challenging task. This paper describes our work to develop an initial model for this process. Based on simulation results, it is determined that this initial model has the potential to be the basis for the development of the complete nonlinear model that can be used for development of a viable control method which offers the likelihood for improvement in the control of the hot metal rolling process.

Simultaneous Piezoelectric Actuator and Sensor Placement Optimization and Control Design of Manipulators with Flexible Links Using SDRE Method

This paper presents a control design for flexible manipulators using piezoelectric actuators bonded on nonprismatic links. The dynamic model of the manipulator is obtained in a closed form through the Lagrange equations. Each link is discretized using finite element modal formulation based on Euler‐Bernoulli beam theory. The control uses the motor torques and piezoelectric actuators for controlling vibrations. An optimization problem with genetic algorithm (GA) is formulated for the location and size of the piezoelectric actuator and sensor on the links. The natural frequencies and mode shapes are computed by the finite element method, and the irregular beam geometry is approximated by piecewise prismatic elements. The State‐Dependent Riccati Equation (SDRE) technique is used to derive a suboptimal controller for a robot control problem. A state‐dependent equation is solved at each new point obtained for the variables from the problem, along the trajectory to obtain a nonlinear feedback controller. Numerical tests verify the efficiency of the proposed optimization and control design.

Rotorcraft Waypoint Guidance Design Using SDRE Controller

This paper deals with the State-Dependent Riccati Equation (SDRE) Technique for the design of rotorcraft waypoint guidance. To generate the flight trajectory through multiple waypoints, we use the trigonometric spline. The controller design and its validation is based upon a level 2 simulation rotorcraft model and the designed SDRE controller is applied to the trajectory tracking problems. To verify the designed guidance law, the simulation environment of high fidelity rotorcraft model is developed using three independent PCs. This paper focuses on the validation of rotorcraft waypoint guidance law which is designed by using SDRE Controller.

SDRE 기법을 이용한 비선형 헬리콥터의 비행 경로점 유도제어

This paper deals with the State-Dependent Riccati Equation (SDRE) Technique for the design of helicopter nonlinear waypoint guidance controller. To generate the flight guidance through multiple waypoints, we use the trigonometric spline. The controller design and its validation is based upon a level 2 simulation helicopter model and the designed SDRE controller is applied to the trajectory tracking problems. To validate the designed SDRE controller, the simulation environment of high fidelity helicopter model is developed using three independent computers. This paper focuses on the validation the present SDRE controller through the helicopter waypoint guidance simulation.

Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique

The goal of the present study is to develop a decentralized coordinated attitude control algorithm for satellite formation flying. To handle the non-linearity of the dynamic system, the problems of absolute and relative attitude dynamics are formulated for the state-dependent Riccati equation (SDRE) technique. The SDRE technique is for the first time utilized as a non-linear controller of the relative attitude control problem for satellite formation flying, and then the results are compared to those from linear control methods, mainly the PD and LQR controllers. The stability region for the SDRE-controlled system was obtained using a numerical method. This estimated stability region demonstrates that the SDRE controller developed in the present paper guarantees the globally asymptotic stability for both the absolute and relative attitude controls. Moreover, in order to complement a non-selective control strategy for relative attitude error in formation flying, a selective control strategy is suggested. This strategy guarantees not only a reduction in unnecessary calculation, but also the mission-failure safety of the attitude control algorithm for satellite formation. The attitude control algorithm of the formation flying was tested in the orbital-reference coordinate system for the sake of applying the control algorithms to Earth-observing missions. The simulation results illustrate that the attitude control algorithm based on the SDRE technique can robustly drive the attitude errors to converge to zero.

Nonlinear optimal integrated vehicle control using individual braking torque and steering angle with on-line control allocation by using state-dependent Riccati equation technique

This paper proposes the solution of state-dependent Riccati equation as a nonlinear optimal regulator to stabilise the motion dynamics of the vehicle model subjected to sudden disturbance inputs in the lateral direction. The proposed nonlinear regulator coordinates individually actuated wheel braking torque and steering wheel angle simultaneously in an optimal manner. Performance criteria are satisfied by solving the Riccati equation based on the given cost function subjected to the nonlinear vehicle dynamics. On-line control allocation in terms of optimal brake torque distribution enhanced by optimal wheel steering angle input is achieved. Furthermore, the proposed optimal nonlinear regulator is an active fault-tolerant control system against partial by-wire actuator failures while guaranteeing stability with good performance due to its capability to allocate the individual control inputs in an optimal way. The main aim is to stabilise the motion dynamics of the vehicle model during short-term emergency situations along the desired straight trajectory manageable by average drivers and to provide vehicle stability and handling predictability through the interaction of individual wheel braking and steering actuators. Simulation results are used to illustrate the effectiveness of the proposed methodology.

Control Strategy for a Snake-Like Robot Based on Constraint Force and Verification by Experiment

This paper describes a locomotion method of a snake-like robot with passive wheels based on a stabilizing control theory for nonlinear systems. The advantage of this method is that the locomotion can be realized only by a state feedback control law, while many reported methods require some reference trajectory such as the serpenoid curve or are realized not only by a locomotion control, but also by a posture control to avoid singular postures. The proposed method evaluates the friction force of passive wheels in the quadratic-like cost function, so that the efficiency with respect to the input can be improved. The state-dependent Riccati equation technique is utilized to realize this method and it permits us to tune the system performance like an optimal control case. The projection method, one of the nice modeling techniques, is utilized to derive both the plant and the friction model. The effectiveness of the proposed method is verified through numerical simulations and experiments.